There are two types of data:

• Qualitative / non-numerical
• Quantitative / numerical :
  • Discrete data
  • Continuous data

Measures of central tendency

• Arithmetic mean:

$$ \left< x \right>=\frac{1}{N}\sum^N_{i=1}x_i $$

• Geometric mean:

$$ GM=\left(\prod^N_{i=1}x_i \right)^\frac{1}{N}=\exp\left[\frac{1}{N}\sum^N_{i=1}\ln(x_i)\right] $$

• Harmonic mean:

$$ H=\frac{N}{\sum^N_{i=1}\frac{1}{x_i}} $$

• Root mean square

$$ \text{RMS}=\sqrt{\frac{\sum^N_{i=1}x_i^2}{N}} $$

• median:

$$ \underbrace{1,2,4,8,16}{5},\overbrace{32}^{\text{median}},\underbrace{64,128,256,512,1024}{5} $$

• Mode:

  • For discrete distributions

  $$ {\color{red}1},{\color{orange}2},{\color{cyan}3},{\color{cyan}3},4,{\color{cyan}3},{\color{orange}2},{\color{orange}2},{\color{cyan}3},{\color{red}1}\qquad \text{more 3's so:} \quad \text{mode}=3 $$

  • For continuous distributions

  $$ \text{max}(\text{PDF}) $$

• Arithmetic mean for weighted data:

$$ \left<x\right>=\frac{\sum^J_{j=1}n_jx_j}{\sum^J_{j=1}n_j}\quad \text{or }\lim_{j\to\infin}:\; \left<x\right>=\frac{\int^{x_\text{max}}{x\text{min}}n(x)x\,\text dx}{\int^{x_\text{max}}{x\text{min}}n(x)\,\text dx} $$

Measures of dispersion

• variance (and standard deviation):

$$ V(x)=\sigma^2=\frac{1}{N}\sum^N_{i=1}(x_i-\underbrace{\mu}_{\text{True mean}})^2 $$

$\mu \ne \left<x\right>$, this is why there is a distinction between population variance $V$ and sample variance $s$

• Corrected/unbiased sample variance:

$$ s^2_\text{corr}=\frac{1}{N-1}\sum^N_{i=1}(x_i-\left< x\right>)^2 $$

• Uncorrected/biased sample variance:

$$ s^2_\text{uncorr}=\frac{1}{N}\sum^N_{i=1}(x_i-\left< x\right>)^2 $$